MECH350 Notes 2010 Lecture4

8/8/2019 MECH350 Notes 2010 Lecture4

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Transfer Function Models

of Dynamical Processes

Process Dynamics and ControlProcess Dynamics and Control


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Transfer FunctionsTransfer Functions

For the DC motor,

the Laplace domain dynamics are given by

To get back to time domain, we must

Specify Laplace domain functions Apply partial fraction expansion

Take Inverse Laplace


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The expression

describes the dynamic behavior of the process explicitly

The Laplace domain function is called the transfer function

between and

Transfer functions are usually represented in Block diagram form


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Transfer FunctionTransfer Function

Heated stirredHeated stirred--tank model (constant flow, )tank model (constant flow, )

Taking the Laplace transform yields:Taking the Laplace transform yields:

or lettingor letting

Transfer functions


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Heated stirred tank exampleHeated stirred tank example

e.g.e.g. The block is called the transfer function relating Q(s) to T(s)

+

++


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Block DiagramsBlock Diagrams

Transfer functions of complex systems can be representedTransfer functions of complex systems can be represented

in block diagram form.in block diagram form.

3 basic arrangements of transfer functions:3 basic arrangements of transfer functions:

1.1. Transfer functions in seriesTransfer functions in series

2.2. Transfer functions in parallelTransfer functions in parallel

3.3. Transfer functions in feedback formTransfer functions in feedback form


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Transfer functions in seriesTransfer functions in series

Overall operation is the multiplication of transfer functionsOverall operation is the multiplication of transfer functions

Resulting overall transfer functionResulting overall transfer function


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Transfer functions in series (two first order systems)Transfer functions in series (two first order systems)

Overall operation is the multiplication of transfer functionsOverall operation is the multiplication of transfer functions

Resulting overall transfer functionResulting overall transfer function


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DC Motor example:DC Motor example:

In terms of angular velocityIn terms of angular velocity

In terms of the angleIn terms of the angle


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Transfer function in parallelTransfer function in parallel

Overall transfer function is the addition of TFs in parallelOverall transfer function is the addition of TFs in parallel

+

+


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Transfer function in parallelTransfer function in parallel

Overall transfer function is the addition of TFs in parallelOverall transfer function is the addition of TFs in parallel

+

+


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Transfer functions in (negative) feedback formTransfer functions in (negative) feedback form

Overall transfer functionOverall transfer function


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Transfer functions in (positive) feedback formTransfer functions in (positive) feedback form

Overall transfer functionOverall transfer function


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ExampleExample


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Example 3.20Example 3.20

A positive feedback loopA positive feedback loop


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Two systems in parallelTwo systems in parallel

Replace byReplace by


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Two systems in parallelTwo systems in parallel


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A negative feedback loopA negative feedback loop


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ExampleExample

Two process in seriesTwo process in series


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Transfer functionsTransfer functions

Transfer functions are generally expressed as a ratio ofTransfer functions are generally expressed as a ratio of

polynomialspolynomials

WhereWhere

The polynomial is called theThe polynomial is called the characteristic polynomialcharacteristic polynomialofof

Roots of are theRoots of are the zeroeszeroes ofof

Roots of are theRoots of are thepolespoles ofof


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Transfer functionTransfer function

Order of underlying ODE is given by degree of

characteristic polynomiale.g. First order processes

Second order processes

Orderof the process is the degree of the characteristic

(denominator) polynomial The relative degree is the difference between the degree of the

denominator polynomial and the degree of the numerator

polynomial


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Steady state behavior of the process obtained form the finalSteady state behavior of the process obtained form the final

value theoremvalue theoreme.g. First order processe.g. First order process

For a unitFor a unit--step input,step input,

From the final value theorem, the ultimate value of isFrom the final value theorem, the ultimate value of is

This implies that the limit exists,This implies that the limit exists, i.e.i.e. that the system is stable.that the system is stable.


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Transfer function is the unit impulse responseTransfer function is the unit impulse response

e.g. First order process,e.g. First order process,

Unit impulse response is given byUnit impulse response is given by

In the time domain,In the time domain,


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Unit impulse response of a 1st order processUnit impulse response of a 1st order process

QuickTime and adecompressor

are needed to see this picture.


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Deviation VariablesDeviation Variables

To remove dependence on initial condition

e.g.

Compute equilibrium condition for a given and

Define deviation variables

Rewrite linear ODE

or

0


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Assume that we start at equilibrium

Transfer functions express extent of deviation from a given

steady-state

Procedure

Find steady-state

Write steady-state equation

Subtract from linear ODE

Define deviation variables and their derivatives if required

Substitute to re-express ODE in terms of deviation variables


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In mechanical systems, the equilibrium is usually selectedIn mechanical systems, the equilibrium is usually selected

as the initial rest positionas the initial rest position Cruise control exampleCruise control example

Suspension system exampleSuspension system example

Satellite systemSatellite system

DC motorDC motor

Using initial condition such that the output is at zero,Using initial condition such that the output is at zero,avoids the need for deviation variablesavoids the need for deviation variables

Initial conditions must be an equilibrium of the systemInitial conditions must be an equilibrium of the system


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Deviation variablesDeviation variables

Example (the ball and beam example)Example (the ball and beam example)

This is a nonlinear system of ordinary differential equationsThis is a nonlinear system of ordinary differential equations

Must be linearized about an equilibrium to obtain a transferMust be linearized about an equilibrium to obtain a transfer

function modelfunction model


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Pendulum examplePendulum example

System equations areSystem equations are nonlinearnonlinearinin

For small perturbation about the vertical positionFor small perturbation about the vertical position , the, the

nonlinearity can approximated (1st order Taylor series expansion)nonlinearity can approximated (1st order Taylor series expansion)

Linearized modelLinearized model

Starting at rest,Starting at rest, , taking the Laplace transform, taking the Laplace transform


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Process ModelingProcess Modeling

Gravity tank

Objectives:Objectives: height of liquid in tank

Fundamental quantity:Fundamental quantity: Mass, momentum

Assumptions:Assumptions:

Outlet flow is driven by head of liquid in the tank

Incompressible flow

Plug flow in outlet pipe

Turbulent flow

h

L

F

Fo


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From mass balance and Newtons law,

Asystem ofsimultaneous ordinary differential equations results

Linear or nonlinear?


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Nonlinear ODEsNonlinear ODEs

Q: If the model of the process is nonlinear, how do we

express it in terms of a transfer function?

A: We have to approximate it by a linear one (i.e.Linearize)

in order to take the Laplace.

f(x0)

f(x)

x

x

f

xx( )0

xx0


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Nonlinear systemsNonlinear systems

First order Taylor series expansion

1. Function of one variable

2. Function of two variables

3. ODEs

f x u f xs usf x u

xx xs

f x u

uu us

s s s s( , ) ( , )( , )

( )( , )

( )} x

x

x

x

f x f xsf x

xx xs

s( ) ( )( )

( )} x

x

( ) ( )( )

( )x f x f xsf xs

xx xs! }

x

x


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TransferTransfer))unctionunction

Procedure to obtain transfer function from nonlinear

process models Find an equilibrium point of the system

Linearize about the equilibrium

Express in terms of deviations variables about the equilibrium

Take Laplace transform

Isolate outputs in Laplace domain

Express effect of inputs in terms of transfer functions


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Ball and beam exampleBall and beam example

Linearize the system of equations about equilibriumLinearize the system of equations about equilibrium

The nonlinear model is given byThe nonlinear model is given by

Linearize (1st order Taylor series expansion about equilibrium)Linearize (1st order Taylor series expansion about equilibrium)


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Linearization gives the linear systemLinearization gives the linear system

Taking Laplace transformTaking Laplace transform



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First Order SystemsFirst Order Systems

First order systems are systems whose dynamics areFirst order systems are systems whose dynamics are

described by the transfer functiondescribed by the transfer function

wherewhere

is the systemsis the systems (steady(steady--state) gainstate) gain

is theis the time constanttime constant

First order systems are the most common behaviourFirst order systems are the most common behaviourencountered in practiceencountered in practice


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ExamplesExamples, Liquid storage

Assume:

Incompressible flow

Outlet flow due to gravity

Balance equation: Total

Flow In

Flow Out


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Balance equation:Balance equation:

Deviation variables about the equilibriumDeviation variables about the equilibrium

Laplace transformLaplace transform

First order system withFirst order system with


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ExamplesExamples: Cruise control

DC Motor


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Liquid Storage Tank

Speed of a car

DC Motor

First order processes are characterized by:

1. Their capacity to store material, momentum

and energy

2. The resistance associated with the flow of

mass, momentum or energy in reaching their

capacity


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Step response of first order process

Step input signal of magnitude M

The ultimate change in is given by


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Step responseStep response




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What do we look for?What do we look for?

Systems Gain: SteadySystems Gain: Steady--State ResponseState Response

Process Time Constant:Process Time Constant:

What do we need?What do we need?

System initially at equilibriumSystem initially at equilibrium

Step input of magnitude MStep input of magnitude M

Measure process gain from new steadyMeasure process gain from new steady--statestate

Measure time constantMeasure time constant

Time Required to Reach

63.2% of final value


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First order systems are also called systems with finiteFirst order systems are also called systems with finite

settling timesettling time The settling time is the time required for the system comesThe settling time is the time required for the system comes

within 5% of the total change and stays 5% for all timeswithin 5% of the total change and stays 5% for all times

Consider the step responseConsider the step response

The overall change isThe overall change is


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Settling timeSettling time



i di d


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Process initially at equilibrium subject to a step ofProcess initially at equilibrium subject to a step of

magnitude 1magnitude 1

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e

Fi dFi d


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First order processFirst order process

Ramp response:Ramp response:

Ramp input of slope a

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fi O d SFi O d S


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Sinusoidal responseSinusoidal response

Sinusoidal inputAsin([t)

0 2 4 6 8 10 12 14 16 18 20 -1.5

-1

-0.5

0

0.5

1

1.5

2

AR

J

Fi O d SFi O d S


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10-2

10-1

100

101

102

10-2

10-1

100 Bode Plots

10-2

10-1

100

101

102

-100

-80

-60

-40

-20

0

High FrequencyAsymptoteCorner Frequency

Amplitude Ratio Phase Shift

I t ti S tI t ti S t


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Integrating SystemsIntegrating Systems

Example: Liquid storage tankExample: Liquid storage tank

Laplace domain dynamicsLaplace domain dynamics

If there is no outlet flow,If there is no outlet flow,

h

F

Fi



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ExampleExample

CapacitorCapacitor

Dynamics of both systems is equivalentDynamics of both systems is equivalent



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Step input of magnitude MStep input of magnitude M

TimeTime

Slope =



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Unit impulse responseUnit impulse response

TimeTime



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Rectangular pulse responseRectangular pulse response

TimeTime

S d d S tS d d S t


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Second order SystemsSecond order Systems

Second order process:Second order process:

Assume the general formAssume the general form

wherewhere = Process steady= Process steady--state gainstate gain

= Process time constant= Process time constant= Damping Coefficient= Damping Coefficient

Three families of processesThree families of processes

UnderdampedUnderdampedCritically DampedCritically Damped

OverdampedOverdamped

Second Order S stemsSecond Order S stems


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Second Order SystemsSecond Order Systems

Three types of second order process:

1. Two First Order Systems in series or in parallel

e.g. Two holding tanks in series

2. Inherently second order processes: Mechanical systemspossessing inertia and subjected to some external force

e.g. A pneumatic valve

3. Processing system with a controller: Presence of a

controller induces oscillatory behaviore.g. Feedback control system



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Multicapacity Second Order ProcessesMulticapacity Second Order Processes

Naturally arise from two first order processes in seriesNaturally arise from two first order processes in series

By multiplicative property of transfer functions

By multiplicative property of transfer functions



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First order systems in parallelFirst order systems in parallel

Overall transfer function a second order process (with one zero)Overall transfer function a second order process (with one zero)

+

+



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1


Inherently second order process:Inherently second order process:

e.g. Pneumatic Valve

x

p

By Newtons law



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Feedback Control SystemsFeedback Control Systems



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Second order process:Second order process:

Assume the general formAssume the general form

wherewhere = Process steady= Process steady--state gainstate gain

= Process time constant= Process time constant= Damping Coefficient= Damping Coefficient

Three families of processesThree families of processes

UnderdampedUnderdampedCritically DampedCritically Damped

OverdampedOverdamped



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Roots of the characteristic polynomial

Case 1) Two distinct real roots

System has an exponential behavior

Case 2) One multiple real root

Exponential behavior

Case 3) Two complex roots

System has an oscillatory behavior



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Step response of magnitude MStep response of magnitude M

0 1 2 3 4 5 6 7 8 9 10 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

\!

\!

\!



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Observations

Responses exhibit overshoot when

Large yield a slow sluggish response

Systems with yield the fastest response without overshoot

As with ) becomes smaller, system becomes more

oscillatory



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Characteristics of underdamped second order process

1. Rise time,

2. Time to first peak,

3. Settling time,

4. Overshoot:

5. Decay ratio:



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Step responseStep response





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Sinusoidal ResponseSinusoidal Response

wherewhere



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Qui i

i i ure.

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MECH350 Notes 2010 Lecture4

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